Factoring multivariate polynomials over the integers

Lower bounds for decomposable univariate wild polynomials

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots of the polynomial and whose lengths are less than the given error bound, The algorithms also provide a simple means of determining the number of real roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 < ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

We present a new approach to the problem of computing the zeta function of a hypersurface over a finite field. For a hypersurface defined by a polynomial of degree d in n variables over the field of q elements, one desires an algorithm whose running time is a polynomial function of dn log(q). (Here we assume d ⩾ 2, for otherwise the problem is easy.) The case n = 1 is related to univariate polynomial factorisation and is comparatively straightforward. When n = 2 one is counting points on curves, and the method of Schoof and Pila yields a complexity of log ⁡ ( q ) C d , where the function Cd depends exponentially on d. For arbitrary n, the theorem of the author and Wan gives a complexity which is a polynomial function of (pdn log(q))n, where p is the characteristic of the field. A complexity estimate of this form can also be achieved for smooth hypersurfaces using the method of Kedlaya, although this has only been worked out in full for curves. The new approach we present should yield a complexity which is a small polynomial function of pdn log(q). In this paper, we work this out in full for Artin–Schreier hypersurfaces defined by equations of the form Zp − Z = f, where the polynomial f has a diagonal leading form. The method utilises a relative p-adic cohomology theory for families of hypersurfaces, due in essence to Dwork. As a corollary of our main theorem, we obtain the following curious result. Let f be a multivariate polynomial with integer coefficients whose leading form is diagonal. There exists an explicit deterministic algorithm which takes as input a prime p, outputs the number of solutions to the congruence equation f = 0 op, and runs in {\mathcal O}(p^{2 + \varepsilon}) bit operations, for any ε > 0. This improves upon the elementary estimate of {\mathcal O}(p^{n-1 + \varepsilon}) bit operations, where n is the number of variables, which can be achieved using Berlekamp's root counting algorithm. 2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y − f 1(X))…(Y − f n (X)) + g(X), with f, f 1, ⋅, f n , g univariate polynomials over an arbitrary field.

Our results concern the Macdonald q, t-Kostka coefficients K λµ(q, t). More precisely we work here with the expressions \(\tilde K_{\lambda \mu } (q,t) = t^{n(\mu )} K_{\lambda \mu } (q,1/t)\) where n(µ) denotes the sum of the legs of the cells of µ. The K λµ (q, t) have been conjectured by Macdonald to be polynomials in q, t with positive integer coefficients. We prove here the Macdonald conjecture for arbitrary µ when λ is an augmented hook. Our proof is based on explicit formulas yielding \(\tilde K_{\lambda \mu } (q,t)\) as a symmetric polynomial plethystically evaluated at \(\tilde K_{(n - 1,1),\mu } (q,t)\).More precisely, it can be shown that for each γ ⊢ k there is a unique symmetric polynomial kγ (x; q, t), of degree ≤ k in x, such that for any λ of the form λ = (n − k, γ), we have \(\tilde K_{(n - 1,1),\mu } (q,t) = k_\lambda \left[ {\tilde K_{(n - 1,1),\mu } (q,t);q,t} \right]\). The proof of existence of the polynomials kγ (x; q, t) is algorithmic. There are now two separate algorithms yielding kγ (x; q, t). We present here the chronologically first one. The second one is presented in a paper by Garsia-Tesler [12] where it yields that kγ (x;q, t) is a polynomial with integer coefficients in x, q, t, 1/q, 1/t. Although the first algorithm is not as good with denominators, we present it here since, combined with some recent work of Lapointe-Vinet, it yields what may be the simplest proof that the K λµ (q, t) are polynomials with integer coefficients.

Congruence arithmetic algorithms for polynomial real zero determination

This paper discusses a set of algorithms which given a univariate polynomial with integer coefficients (with possible multiple zeros) and a positive rational error bound, uses infinite-precision integer arithmetic and Sturm's Theorem to compute intervals containing the real zeros of the polynomial and whose lengths are less than the given error bound. The algorithms also provide a simple means of determining the number of real zeros in any interval. Theoretical computing time bounds are developed for the algorithms and some empirical results are reported.

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = Q(α1,..., αe) is an algebraic extension of the rational numbers. Our bounds are single exponential in e and match the ones presented in [34] for the case e = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is OB(N4e+4), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of OB [equation], respectively. We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets.

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if n has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs.We present a normal form for collisions of any number of decompositions with any number of components and describe exactly the (non)uniqueness of the parameters in the tame case. This yields an exact formula for the number of such collisions over a finite field. We conclude with a fast algorithm for the exact number of decomposable polynomials of degree n over a finite field Fq of characteristic coprime to n.

Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks.Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales poly-logarithmically with it, have been known to exist thus far only over a set of finite fields of negligible density, namely, over “FFT-friendly” fields that contain a sub-group of size \(2^\mathsf{{k}} \).Our main result is to show that scalable IOPs can be constructed over any sufficiently large finite field, of size that is at least quadratic in the length of computation whose integrity is proved by the IOP. This result has practical applications as well, because it reduces the proving and verification complexity of cryptographic statements that are naturally stated over pre-defined finite fields which are not “FFT-friendly”. Prior state-of-the-art scalable IOPs relied heavily on arithmetization via univariate polynomials and Reed–Solomon codes over FFT-friendly fields. To prove our main result and extend scalability to all large finite fields, we generalize the prior techniques and use new algebraic geometry codes evaluated on sub-groups of elliptic curves (elliptic curve codes). We also show a new arithmetization scheme that uses the rich and well-understood group structure of elliptic curves to reduce statements of computational integrity to other statements about the proximity of functions evaluated on the elliptic curve to the new family of elliptic curve codes.

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [11] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is to put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.

Let \({\mathcal{O}}\) be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of \({\mathcal{O}}\) a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

Division algorithms for Bernstein polynomials

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

Compositions and collisions at degree [formula omitted